SME
Kinematics of Mechanisms
Gear trains
Use this for “A 15-tooth pinion drives a 45-tooth gear keyed to a 24-tooth pinion, which drives a 48-tooth gear. The input turns at 1500 rpm. Find the train value and output speed.”
Diagram
Gears
2
and
3
share
a
shaft.
e
=
ΠN
driving
ΠN
driven
=
15
×
24
45 × 48
=
0.167.
n
out
=
e
×
n
in
=
250
rpm,
the
same
way
round
as
gear
1.
This is an example. Type your own values to start a new problem.
e
=
×
×
nₒᵤₜ (rpm)
=
e
×
nᵢₙ (rpm)
Formulas
e
=
N₁
×
N₃
N₂
×
N₄
0.1667
=
15 × 24
45 × 48
nₒᵤₜ
=
e
×
nᵢₙ
250 = 0.1667 × 1,500
Assumptions
•
Gears 2 and 3 are keyed to one shaft, so they turn together.
•
Gears 1 and 3 drive; gears 2 and 4 are driven.
•
Each external mesh reverses the turning, so after two meshes the output turns the input’s way.
•
The teeth roll without slip, so the speeds follow the tooth counts.
Step-by-step
WE KNOW
N₁ = 15, N₂ = 45, N₃ = 24, N₄ = 48, nᵢₙ = 1,500 rpm
FIND
train value (e), output speed (nₒᵤₜ)
Step 1 · Find train value (e)
Use e = (N₁ × N₃) ÷ (N₂ × N₄)
Each mesh scales the speed by driving teeth over driven teeth; the two ratios multiply.
e
=
15 × 24
45 × 48
e
=
15
×
24
2,160
e
=
360
2,160
e
=
0.1667
Step 2 · Find output speed (nₒᵤₜ)
Use nₒᵤₜ = e × nᵢₙ
The
output
turns
e
times
as
fast
as
the
input.
nₒᵤₜ = 0.1667 × 1,500
nₒᵤₜ = 250 rpm
Check
Put the numbers back in. Both sides should match.
0.1667
=
15 × 24
45 × 48
✓
250 = 0.1667 × 1,500 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
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Torque and power through a train
Use this for “A gearbox takes 50 N·m at 1800 rpm and turns its output at 200 rpm with 95% efficiency. Find the output torque and the power.”